Controlling cascade dressing interaction of four-wave mixing image

doi:10.1364/OE.19.013675

Controlling cascade dressing interaction of four-wave mixing image

Changbiao Li, Yanpeng Zhang, Huaibin Zheng, Zhiguo Wang, Haixia Chen, Suling Sang, Ruyi Zhang, Zhenkun Wu, Liang Li, and Peiying Li »View Author Affiliations

Optics Express, Vol. 19, Issue 14, pp. 13675-13685 (2011)
http://dx.doi.org/10.1364/OE.19.013675

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▸ Article Outline
– Abstract
1. Introduction
2. Theoretical model and experimental scheme
3. Cascade dressing interaction
4. Conclusion
– References and links

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Abstract

We report our observations on enhancement and suppression of spatial four-wave mixing (FWM) images and the interplay of four coexisting FWM processes in a two-level atomic system associating with three-level atomic system as comparison. The phenomenon of spatial splitting of the FWM signal has been observed in both x and y directions. Such FWM spatial splitting is induced by the enhanced cross-Kerr nonlinearity due to atomic coherence. The intensity of the spatial FWM signal can be controlled by an additional dressing field. Studies on such controllable beam splitting can be very useful in understanding spatial soliton formation and interactions, and in applications of spatial signal processing.

1. Introduction

Efficient four-wave mixing (FWM) processes enhanced by atomic coherence in multilevel atomic systems [1

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–9999 (1997). [CrossRef]

–4

H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80(2), 023820 (2009). [CrossRef]

] are of great current interest. Recently, destructive and constructive interferences in a two-level atomic system [5

S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98(5), 053601 (2007). [CrossRef] [PubMed]

] and competition via atomic coherence in a four-level atomic system [6

Y. Zhang, B. Anderson, A. W. Brown, and M. Xiao, “Competition between two four-wave mixing channels via atomic coherence,” Appl. Phys. Lett. 91(6), 061113 (2007). [CrossRef]

] with two coexisting FWM processes were studied. Also, the interactions of doubly dressed states and the corresponding effects of atomic systems have attracted many researchers in recent years [7

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]

M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A 64(1), 013412 (2001). [CrossRef]

]. The interaction of double-dark state and splitting of a dark state in a four-level atomic system were studied theoretically in an electromagnetically induced transparency (EIT) system by Lukin et al. [7

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]

]. The triple-peak absorption spectrum, which was observed later in the N-type cold atomic system by Zhu et al., verified the existence of the secondarily dressed states [8

M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A 64(1), 013412 (2001). [CrossRef]

]. Recently, we had theoretically investigated three types of doubly dressed schemes in a five-level atomic system [9

Z. Q. Nie, H. B. Zheng, P. Z. Li, Y. M. Yang, Y. P. Zhang, and M. Xiao, “Interacting multi-wave mixing in a five-level atomic system,” Phys. Rev. A 77(6), 063829 (2008). [CrossRef]

] and observed three-peak Autler-Townes (AT) splitting of the secondary dressing FWM signal [10

Y. P. Zhang, Z. Q. Nie, Z. G. Wang, C. B. Li, F. Wen, and M. Xiao, “Evidence of Autler-Townes splitting in high-order nonlinear processes,” Opt. Lett. 35(20), 3420–3422 (2010). [CrossRef] [PubMed]

]. In addition, we reported the evolution of suppression and enhancement of FWM signal by controlling an additional laser field [11

C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. 95(4), 041103 (2009). [CrossRef]

As two or more laser beams pass through an atomic medium, the cross-phase modulation (XPM), as well as modified self-phase modulation (SPM), can potentially affect the propagation and spatial patterns of the incident laser beams. Laser beam self-focusing [12

G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. 64(21), 2487–2490 (1990). [CrossRef] [PubMed]

] and pattern formation [13

R. S. Bennink, V. Wong, A. M. Marino, D. L. Aronstein, R. W. Boyd, C. R. Stroud Jr, S. Lukishova, and D. J. Gauthier, “Honeycomb pattern formation by laser-beam filamentation in atomic sodium vapor,” Phys. Rev. Lett. 88(11), 113901 (2002). [CrossRef] [PubMed]

] have been extensively investigated with two laser beams propagating in atomic vapors. Recently, we have observed spatial shift [14

A. J. Stentz, M. Kauranen, J. J. Maki, G. P. Agrawal, and R. W. Boyd, “Induced focusing and spatial wave breaking from cross-phase modulation in a self-defocusing medium,” Opt. Lett. 17(1), 19–21 (1992). [CrossRef] [PubMed]

] and spatial splitting [15

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87(7), 073601 (2001). [CrossRef] [PubMed]

–17

Y. P. Zhang, Z. G. Wang, H. B. Zheng, C. Z. Yuan, C. B. Li, K. Q. Lu, and M. Xiao, “Four-wave-mixing gap solitons,” Phys. Rev. A 82(5), 053837 (2010). [CrossRef]

] of the FWM beams generated in multi-level atomic systems, which can be well controlled by additional dressing laser beams via XPM. Studies on such spatial shift and splitting of the laser beams can be very useful in understanding the formation and interactions of spatial solitons [16

W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous interaction of spatial solitons in photorefractive media,” Phys. Rev. Lett. 80(15), 3240–3243 (1998). [CrossRef]

] in the Kerr nonlinear systems and signal processing applications, such as spatial image storage [18

P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, “Storage and retrieval of multimode transverse images in hot atomic Rubidium vapor,” Phys. Rev. Lett. 100(12), 123903 (2008). [CrossRef] [PubMed]

], entangled spatial images [19

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008). [CrossRef] [PubMed]

], soliton pair generation [20

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. L. Luther-Davies, “Observation of dipole-mode vector solitons,” Phys. Rev. Lett. 85(7), 1424–1427 (2000). [CrossRef] [PubMed]

], and influences of higher-order (such as fifth-order) nonlinearities [21

Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four- wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102(1), 013601 (2009). [CrossRef] [PubMed]

In this paper, we first report our experimental studies of the interaction of four coexisting FWM processes in a two-level atomic system by blocking different laser beams. Next, we investigate the various suppression/enhancement of the degenerate-FWM (DFWM) signals and two dispersion centers, which are caused by the cascade dressing interaction of two dressing fields. The experimental results clearly show the evolutions of the enhancement and suppression, from pure enhancement to partial enhancement/suppression, then to pure suppression, further to partial enhancement/suppression, and finally to enhancement, which are in good agreement with the theoretical calculations. In addition, we also observe the spatial splitting in the x and y directions of DFWM signal due to different spatially alignment of the probe and coupling beams.

2. Theoretical model and experimental scheme

The two relevant experimental systems are shown in Figs. 1(a) and 1(b). Three energy levels from sodium atom in heat pipe oven are involved in the experimental schemes. The pulse laser beams are aligned spatially as shown in Fig. 1(c). In the Fig. 1(a), energy levels |0〉 (

3 S_{1 / 2}

) and |1〉 (

3 P_{3 / 2}

) form a two-level atomic system. Coupling field

E_{1}

(with wave vector

k_{1}

and the Rabi frequency

G_{1}

) together with

E_{1}^{'}

(

k_{1}^{'}

and

G_{1}^{'}

) (connecting the transition between |0〉 and |1〉) having a small angle (

\sim {0.3}^{\circ}

) propagates in the opposite direction of the probe field

E_{3}

(

k_{3}

and

G_{3}

) (also connecting the transition between |0〉 and |1〉). These three laser beams come from the same near-transform-limited dye laser (with a 10 Hz repetition rate, 5 ns pulse width, and

0.04

{cm}^{- 1}

linewidth) with the same frequency detuning

Δ_{1} = ω_{10} - ω_{1}

, where

ω_{10}

is the transition frequency between |0〉 and |1〉. The coupling fields

E_{1}

and

E_{1}^{'}

induce a population grating between states |0〉 to |1〉, which is probed by

E_{3}

. This generates a DFWM process (Fig. 1(a)) satisfying the phase-matching condition of

k_{F 1} = k_{1} - k_{1}^{'} + k_{3}

. Then, two additional coupling fields

E_{2}

(

k_{2}

G_{2}

) and

E_{2}^{'}

(

k_{2}^{'}

G_{2}^{'}

) are applied as scanning fields connecting the transition from |0〉 to |1〉 with the same frequency detuning

Δ_{2} = ω_{10} - ω_{2}

; the two additional coupling fields are from another similar dye laser set at

ω_{2}

to dress the energy level |1〉. The fields

E_{2}

E_{2}^{'}

, and

E_{3}

produce a non-degenerate FWM (NDFWM) signal

k_{F 2}

(satisfying

k_{F 2} = k_{2} - k_{2}^{'} + k_{3}

). When the five laser beams are all on, there also exist other two FWM processes

k_{F 3}

(satisfying

k_{F 3} = k_{2} - k_{1}^{'} + k_{3}

) and

k_{F 4}

(satisfying

k_{F 4} = k_{1} - k_{2}^{'} + k_{3}

) in the same directions as

E_{F 1}

and

E_{F2}

, respectively.

Fig. 1 (a) and (b) The diagram of relevant Na energy levels. (c) The scheme of the experiment. Inset gives the spatial alignments of the incident beams.

Under the experimental condition,

E_{1}

(or

E_{1}^{'}

) with detuning

Δ_{1}

depletes two groups of atoms with different velocities at the same time, such as negative velocities group and positive velocities group. At

Δ_{1} < 0

, the positive velocities group will see

E_{1}

(or

E_{1}^{'}

) with detuning

Δ_{1} + k_{1} v

and

E_{3}

with detuning

Δ_{1} - k_{3} v

. The frequency of the DFWM

E_{F 1}

in this case will be

ω_{f} = (ω_{1} - k v) - (ω_{1} - k v) + (ω_{1} + k_{3} v) = ω_{1} + k_{3} v

due to the conservation of energy. Correspondingly, at

Δ_{1} > 0

, negative velocities group will see

E_{1}

(or

E_{1}^{'}

) with detuning

Δ_{1} - k_{1} v

and

E_{3}

with detuning

Δ_{1} + k_{3} v

. The frequency of

E_{F 1}

will be

ω_{f} = (ω_{1} + k v) - (ω_{1} + k v) + (ω_{1} - k_{3} v) = ω_{1} - k_{3} v

. Such changing implies that a group of atoms with certain velocities can satisfy the condition

{Δ^{'}}_{1} = Δ_{1} \pm k_{1} v

, where

{Δ^{'}}_{1}

is the detuning of

E_{1}

(or

E_{1}^{'}

) based on both saturation excitation and atomic coherence effect. As a result, the self-dressing field

E_{1}

(or

E_{1}^{'}

) can be considered as the outer dressing field and separates the level |0〉 into two dressing states

| G_{1} \pm >

, as shown in Fig. 2 . In addition, the Doppler effect and the power broadening effect on the weak FWM signals need to be considered.

Fig. 2 (Color online) The interplay and mutual suppression/enhancement between two coexisting FWM signals (

E_{F 1}

and

E_{F 3}

). (a1) the upper curves: pure DFWM signal

E_{F 1}

(with both

E_{2}

and

E_{2}^{'}

blocked) (squares), singly dressed DFWM signal

E_{F 1}

(with

E_{2}

blocked) (triangles), coexisting singly dressed DFWM signal

E_{F 1}

and FWM signal

E_{F 3}

(with

E_{2}^{'}

blocked) (reverse triangles), and coexisting dressed DFWM signal

E_{F 1}

and FWM signal

E_{F 3}

(circles);lower curves: pure FWM signal

E_{F 3}

(with both

E_{1}

and

E_{2}^{'}

blocked) (left triangle), singly-dressed FWM signal (with

E_{1}

blocked) (right triangle);

Δ_{1} = 0

. The inserted plot: corresponding to the dressed-state picture. (b1) the condition are the same to that in (a1) except

Δ_{1} = - 25.9 GHz

. The inserted plot: corresponding to the dressed-state picture. (a2) and (b2) theoretical plots corresponding to the experimental parameters of (a1) and (b1), respectively.

When

E_{1}

E_{1}^{'}

E_{2}

E_{2}^{'}

and

E_{3}

are open, the DFWM process

E_{F 1}

and NDFWM processes

E_{F 2}

E_{F 3}

and

E_{F 4}

are generated simultaneously, and there exists interplay among these four FWM signals in the two-level atomic system. These generated FWM signals have the frequencies

ω_{F 1} = ω_{F 2} = ω_{1}

ω_{F 3} = ω_{2}

, and

ω_{F 4} = 2 ω_{1} - ω_{2}

. They are split into two equal components by a

50 %

beam splitter before being detected. One is captured by the CCD camera, and the other is detected by photomultiplier tubes (

D_{1}

D_{2}

) and a fast gated integrator (gate width of 50 ns). Also, they are monitored by digital acquisition card.

In order to interpret the following experimental results, we perform the theoretical calculation on the four coexisting FWM processes. First, we consider four FWM processes to be or perturbed by corresponding laser beams. In two-level configuration, there exist the transition paths to generate FWM signals. They can be described by perturbation chains (F1)

ρ_{00}^{(0)} \overset{(E_{1}) *}{\to} ρ_{10}^{(1)} \overset{(E_{1}') *}{\to} ρ_{00}^{(2)} \overset{E_{3}}{\to} ρ_{10}^{(3)}

, (F2)

ρ_{00}^{(0)} \overset{(E_{2}') *}{\to} ρ_{10}^{(1)} \overset{E_{2}}{\to} ρ_{00}^{(2)}

\overset{E_{3}}{\to} ρ_{10}^{(3)}

, (F3)

ρ_{00}^{(0)} \overset{E_{2}}{\to} ρ_{10}^{(1)} \overset{(E_{1}') *}{\to} ρ_{00}^{(2)} \overset{E_{3}}{\to} ρ_{10}^{(3)}

, and (F4)

ρ_{00}^{(0)} \overset{E_{1}}{\to} ρ_{10}^{(1)}

\overset{(E_{2}') *}{\to} ρ_{00}^{(2)} \overset{E_{3}}{\to} ρ_{10}^{(3)}

, respectively. For the DFWM signal

E_{F 1}

, in fact, this DFWM generation process can be viewed as a series of transitions: the first step is from |0〉 to |1〉 with absorption of a coupling photon

E_{1}

, and the final state of this process can be dressed by the dressing field

E_{2}

(or

E_{2}^{'}

). The second step is the transition from |1〉 to |0〉, and the final state cannot be dressed by any field. The third step is the transition from |0〉 to |1〉 with the emission of a probe photon

E_{3}

, and the final state of this process can be dressed by

E_{2}

(or

E_{2}^{'}

). Then, the last transition is from |1〉 to |0〉, which emits a FWM photon at frequency

ω_{1}

. Thus, we can obtain the dressed perturbation chain

ρ_{00}^{(0)} \overset{(E_{1}) *}{\to} ρ_{G 2 \pm 0}^{(1)} \overset{(E_{1}') *}{\to}

ρ_{00}^{(2)} \overset{E_{3}}{\to} ρ_{G 2 \pm 0}^{(3)}

. Similarly, we can obtain the other dressed perturbation chains as (DF2)

ρ_{00}^{(0)} \overset{E_{2}}{\to}

ρ_{G 1 \pm 0}^{(1)} \overset{(E_{2}') *}{\to} ρ_{00}^{(2)} \overset{E_{3}}{\to} ρ_{G 1 \pm 0}^{(3)}

, (DF3)

ρ_{00}^{(0)} \overset{E_{2}}{\to} ρ_{G 2 \pm 0}^{(1)} \overset{(E_{1}') *}{\to} ρ_{00}^{(2)} \overset{E_{3}}{\to} ρ_{G 2 \pm 0}^{(3)}

, and (DF4)

ρ_{00}^{(0)} \overset{E_{1}}{\to}

ρ_{G 1 \pm 0}^{(1)} \overset{(E_{2}') *}{\to}

ρ_{00}^{(2)} \overset{E_{3}}{\to} ρ_{G 1 \pm 0}^{(3)}

, respectively. The expressions of the corresponding density matrix elements related to the four FWM processes are

ρ_{F 1}^{(3)} = - i G_{3} G_{1} {(G_{1}^{'})}^{*} / (Γ_{00} B_{1}^{2})

ρ_{F 2}^{(3)} = - i G_{3} G_{2} {(G_{2}^{'})}^{*} / (Γ_{00} d_{3} B_{2})

ρ_{F 3}^{(3)} = - i G_{2} G_{3} {(G_{1}^{'})}^{*} / (d_{4} B_{3}^{2})

ρ_{F 4}^{(3)} = - i G_{1} G_{3} {(G_{2}^{'})}^{*} / (d_{5} d_{6} B_{1})

, respectively, where

d_{1} = Γ_{10} + i Δ_{1}

d_{2} = Γ_{00} + i (Δ_{1} / m - Δ_{2})

d_{3} = Γ_{10} + i Δ_{2}

d_{4} = Γ_{00} + i (Δ_{2} - Δ_{1})

d_{5} = Γ_{10} + i (2 Δ_{1} - Δ_{2})

d_{6} = [Γ_{00} + i (Δ_{1} - Δ_{2})]

A_{1} = | G_{2} |^{2} / d_{2}

A_{2} = G_{1}^{2} / Γ_{00}

A_{3} = | G_{2} |^{2} / Γ_{00}

B_{1} = d_{1} + A_{1}

B_{2} = d_{1} + A_{2}

B_{3} = d_{3} + A_{3}

. Here

G_{i} = - μ_{i} E_{i} / ℏ

(i = 1, 2, 3) is the Rabi frequency;

Γ_{10}

Γ_{20}

, and

Γ_{00}

are the transverse relaxation rates, and

Δ_{i}

(

i = 1, 2

) is the detuning factor.

The experiments are carried out in a vapor cell containing sodium. The cell, 18-cm long, is heated up to a temperature of about

230^{\circ} C

and crossed by linearly polarized laser beams which interact with the atoms. In the two-level atomic system, the coupling fields

E_{1}

and

E_{1}^{'}

(with diameter of 0.8 mm and power of 9 μW), and the probe field

E_{3}

(with diameter of 0.8 mm and power of 3 μW) are tuned to the line center (589.0 nm) of the

| 0 〉

| 1 〉

transition, which generate the DFWM signal

E_{F 1}

at frequency

ω_{1}

. The coupling fields

E_{2}

and

E_{2}^{'}

(with diameter of 1.1 mm and powers of 20 μW and 100 μW, respectively) are scanned simultaneously around the

| 0 〉

| 1 〉

transition to dress the DFWM process

E_{F 1}

3. Cascade dressing interaction

We first investigate the interaction of four coexisting FWM signals in the two-level atomic system by blocking different laser beams. Firstly, by blocking

E_{2}

(or

E_{1}

), the DFWM signal

E_{F 1}

(or the FWM signal

E_{F 3}

) is suppressed by the coupling field

E_{2}^{'}

as can be seen from the upper triangle points [or the right triangle points in Fig. 2(a1)], compared to the pure DFWM signal

E_{F 1}^{}

(or the FWM signal

E_{F 3}

). Next, when laser beams

E_{1}

E_{1}^{'}

E_{2}

and

E_{3}

are turned on, two coexisting FWM processes (

E_{F 1}

and

E_{F 3}

) couple to each other (the lower triangle points), and the intensities of total FWM signals are increased, as can be attributed to the combination of two FWM signal processes (

E_{F 1}

and

E_{F 3}

). Finally, when all the five laser beams are turned on, the DFWM signal

E_{F 1}

and the FWM signal

E_{F 3}

are both greatly suppressed by corresponding dressing fields. So the intensities of total FWM signals are extremely decreased, as shown in the circles points in Fig. 2(a1).

These effects can be explained effectively by the dressed-state picture. The dressing field

E_{2}^{'}

couples the transition

| 0 >

| 1 >

and creates the dressed states

| G_{2} \pm >

, which leads to single-photon transition

| 0 > \to | 1 >

off-resonance [the inserted plot in Fig. 2(a1)]. At exact single-photon resonance with

Δ_{1} = 0

, the DFWM signal

E_{F 1}

intensity is greatly suppressed by the means of scanning the dressing field

E_{2}^{'}

across the resonance (

Δ_{2} = 0

), as the upper triangle points in Fig. 2(a1) shows. At the same time, the FWM signal

E_{F 3}

experiences similar process [the right triangle points in Fig. 2(a1)].

Furthermore, an appropriate

Δ_{1}

value at which

E_{F 1}

is either enhanced or suppressed is chosen in the investigation. In this case, compared to the pure DFWM signal

E_{F 1}

[square points in Fig. 2(b1)], the dressed DFWM signal

E_{F 1}

is enhanced [the upper and low triangle points in Fig. 2(b1)]. However, the dressed FWM signal

E_{F 3}

is suppressed due to the destructive interference [right triangle points in Fig. 2(b1)] compared to the pure signal [left triangle points in Fig. 2(b1)]. The upper triangle in Fig. 2(b1) combines the two FWM processes (

E_{F 1}

and

E_{F 3}

), which are dressed by laser beams

E_{1}^{'}

and

E_{2}^{'}

, respectively. After calculating

ρ_{F 1}^{(3)}

and

ρ_{F 3}^{(3)}

under the above experiment conditions, good agreements are obtained between the theoretical calculations and the experimental results as shown in Figs. 2(a2) and 2(b2), respectively.

After that, we investigate the evolutions of the interaction between these two coexisting FWM signals by the means of setting different frequency detuning

Δ_{1}

values, where the fixed spectra corresponds to the suppression and enhancement of DFWM signal

E_{F 1}

, and the shifting spectra corresponds to the FWM signal

E_{F 3}

as shown in Figs. 3 (a1)–3(a7). It is obvious in Figs. 3(a1)–3(a3) that, as the frequency detuning

Δ_{1}

varies from

Δ_{1} < 0

to zero from up to down, the DFWM signal

E_{F 1}

shows the evolution from enhancement to partial enhancement/suppression, and then to suppression. At the same time, the FWM signal

E_{F 3}

varies from intense to weak (when two FWM signals

E_{F 1}

and

E_{F 3}

overlap) and shifts from left side to right side, which satisfies the two-photon resonant condition (

Δ_{1} - Δ_{2} = 0

). When

Δ_{1}

changes further to be positive, a symmetric process is observed [i.e., suppression in Fig. 3(a5), partial suppression/enhancement in Fig. 3(a6), and pure enhancement in Fig. 3(a7)]. It should be noted here that FWM signal

E_{F 3}

still shifts from left side to right side. Figure 3(a4) shows the weakened FWM signal due to the strong effect of the Doppler absorption. Especially, the DFWM signal

E_{F 1}

at a large one-photon detuning is extremely weak when

G_{2} = 0

. However, the strong dressing field can cause the resonant excitation of one of the dressed states if the enhanced condition

Δ_{1} - Δ_{2} \pm G_{2} = 0

is satisfied. In such case, the DFWM signal

E_{F 1}

is strongly enhanced [Fig. 3(a1)], mainly due to the one-photon resonance (

| 0 > \to | G_{2} + |

) [the insert plot in Fig. 2(b1)]. As

Δ_{1} = 0

, the intensity of the DFWM signal

E_{F 1}

is greatly suppressed [Fig. 3(a3)], similar to the case of the upper triangle curves in Fig. 2(a1). Also, we can observe the FWM signal

E_{F 3}

is suppressed due to the destructive interference. In addition, Figs. 3(b1)–3(b7) show the interaction of another two coexisting the FWM processes (

E_{F 2}

and

E_{F 4}

), in which the fixed spectra corresponds to the FWM signal

E_{F 2}

, and the shifting spectra corresponds to FWM signal

E_{F 4}

for different frequency detuning

Δ_{1}

values.

Fig. 3 (a) and (b) Measured evolution of the four FWM signals [(

E_{F 1}

and

E_{F 3}

) and (

E_{F 2}

and

E_{F 4}

), respectively] versus

Δ_{2}

for different

Δ_{1}

values. (a1)-(a7) and (b1)-(b7):

Δ_{1} =

-139.1, −103.87, −29.5, 0, 29.8, 100.1, 155.7GHz, respectively.

Now, we concentrate on the cascade dressing interaction and the two dispersion centers of FWM images with two dressing fields

E_{1}^{'}

and

E_{2}^{'}

in the two-level atomic system. In order to investigate the cascade dressing interaction, the power of the coupling field

E_{1}^{'}

is set at

80 μ W

. So the DFWM signal

E_{F 1}

shows a spectrum of the AT splitting due to self-dressed effect [10

] induced by beam

k_{1}^{'}

when

Δ_{1}

is scanned and the dressing field

E_{2}^{'}

is off, as shown in the dashed curve of Fig. 4(a) . When the beam

k_{2}^{'}

is on, the DFWM signal

E_{F 1}

is dressed by both

E_{1}^{'}

and

E_{2}^{'}

, and therefore shows the cascade dressing interaction, as shown in Figs. 4(a) and 4(c). Specifically, by discretely choosing different detuning values within

Δ_{1} < 0

and scanning

Δ_{2}

, the DFWM signal

E_{F 1}

shows the evolution of the successively occurring pure enhancement, partial suppression/enhancement, pure suppression, partial enhancement/suppression and enhancement processes, as shown in the left side of Fig. 4(a). When

Δ_{1}

changes to be positive, a symmetric process occurs, in the right side of Fig. 4(a), which is well described by the theoretical curves [Fig. 4(b)].

Fig. 4 (Color online) (a) Measured suppression and enhancement of DFWM signal

E_{F 1}

versus

Δ_{2}

for different

Δ_{1}

values in the two-level system.

Δ_{1} =

-69.1, −55.5, −38.7, −19.2, 0, 14.7, 28.8, 42.2 and 57.3 GHz, respectively. The dashed curve is the double-peak DFWM signal

E_{F 1}

versus

Δ_{1}

. (b) Theoretical plots corresponding to the experimental parameters in (a). (c) The same measures to (a) with the same condition, except that the laser beams

E_{1}

E_{1}^{'}

overlap in the middle of heat oven. (d1)-(d9) the dressed-state pictures of the suppression or enhancement of the DFWM signal. The states

| G_{1} \pm >

(dashed lines) and the states

| G_{1} + \pm >

| G_{1} - \pm >

(dot-dashed lines), respectively.

In order to explain this phenomenon, the dressed-state picture is adopted, as shown in Fig. 4(d). First, the DFWM signal

E_{F 1}

is dressed by both fields

E_{2}^{'}

and

E_{1}^{'}

. The corresponding expression of the modified density matrix element of DFWM

E_{F 1}

process is

{ρ^{'}}_{E_{F 1}}^{(3)} = - i G_{3} G_{1} {(G_{1}^{'})}^{*} / (B_{4} B_{5}^{2})

, where

d_{6} = Γ_{00} + A_{5}

d_{7} = Γ_{10} - i Δ_{2}

A_{4} = G_{1}^{2} / B_{1}

A_{5} = G_{2}^{2} / d_{7}

A_{6} = G_{1}^{2} / d_{6}

B_{4} = Γ_{00} + A_{4}

and

B_{5} = B_{1} + A_{6}

. Next, the inner dressing field

E_{1}^{'}

dresses the state

| 0 >

to create two new dressing states

| G_{1} \pm >

, and then the strong dressing field

E_{2}^{'}

creates new states

| G_{1} + \pm >

| G_{1} - \pm >

around states

| G_{1} \pm >

, as scanning the frequency detuning

Δ_{2}^{}

. As a result of this dressing scheme, the DFWM signal

E_{F 1}

is extremely small when

G_{2} = 0

and

Δ_{1}

is set far away from the resonance point at both

Δ_{1} < 0

and

Δ_{1} > 0

, respectively. Another result is that the strong fields can cause resonant excitation of one of the dressed state (i.e.,

| G_{1} - - >

| G_{1} + + >

[Fig. 4(d1) and Fig. 4(d9), respectively], which can lead to the enhancement of FWM signals. Specifically, if the condition

Δ_{1} - Δ_{2} = - G_{2}^{}

and

Δ_{1} - Δ_{2} = G_{2}^{}

[corresponding to the dressed states shown in Figs. 4(d1) and 4(d9)] is satisfied, the DFWM signal

E_{F 1}

is obviously enhanced, as shown in the curves of Fig. 4(a1) and 4(a9), respectively. As

Δ_{1}

changes to be near the resonance point, we can get a partial enhancement/suppression of DFWM signal

E_{F 1}

. The first and second transition states satisfy suppression condition

Δ_{1} - Δ_{2} = 0

and enhancement condition

Δ_{1} - Δ_{2} = - G_{2}^{}

(Fig. 4(d2)), [enhancement condition

Δ_{1} - Δ_{2} = G_{2}^{}

and suppression condition

Δ_{1} - Δ_{2} = 0

, as shown in Fig. 4(d4)], as leads to the first suppression and next enhancement (the curve of Fig. 4(a2)) [or the first enhancement and next suppression, as shown in the curve of Fig. 4(a4)]. When

Δ_{1}

reaches the point

Δ_{1} - Δ_{2} = 0

, the suppression effect gets dominant due to the dressed states

| G_{1} - - >

[Fig. 4(d3)], so the DFWM signal

E_{F 1}

is purely suppressed, as shown in the curve of Fig. 4(a3). For the point

Δ_{1}^{} = Δ_{2}^{} = 0

between the two resonance points, the curve of Fig. 4(a5) shows a pair of suppressed peaks. In fact, they are induced by the outer dressing field

E_{2}^{'}

which can largely weaken the suppression effect of the inner dressing field

E_{1}^{'}

on DFWM signal. Furthermore, the other cascade dressing field

E_{2}^{'}

splits such suppressed peak into a pair of suppressed peaks, as shown in the curve of Fig. 4(a5). Figure 4(a) shows the various suppression/enhancement of the DFWM signal

E_{F 1}

and its two dispersion centers, which is caused by the cascade dressing interaction of the two dressing fields

E_{1}

and

E_{2}^{'}

In addition, the spatial splitting in x-direction of the FWM signal beams induced by additional dressing laser beams is observed simultaneously as shown in Fig. 5(a) . It is observed that, the number of the splitting spots increases when the FWM intensity is suppressed. To understand these phenomena, we need to consider the cross-phase modulation (XPM) on the FWM signals. As described in our previous investigation [17

Y. P. Zhang, Z. G. Wang, H. B. Zheng, C. Z. Yuan, C. B. Li, K. Q. Lu, and M. Xiao, “Four-wave-mixing gap solitons,” Phys. Rev. A 82(5), 053837 (2010). [CrossRef]

], the spatial splitting of the FWM beam can be controlled by the intensities of the involved laser beams, the cross-Kerr nonlinear coefficients and the atomic density, according to the nonlinear phase shift

ϕ = 2 k_{F 1} n_{2} z I_{1} e^{- r^{2} / 2} / (n_{0} I_{F 1}^{'})

. Here the additional transverse propagation wave-vector is

δ k_{r} = \partial ϕ / \partial r

. The change of phase (ϕ) distribution in the laser propagating equations determines the spatial splitting of the laser beams. In theoretical calculation, we can obtain the intensity of the

E_{F 1}

beam by

I_{F 1}^{'} \propto | {ρ^{'}}_{E_{F 1}}^{(3)} |^{2} = | - i G_{3} F_{a}^{} |^{2}

[17

Y. P. Zhang, Z. G. Wang, H. B. Zheng, C. Z. Yuan, C. B. Li, K. Q. Lu, and M. Xiao, “Four-wave-mixing gap solitons,” Phys. Rev. A 82(5), 053837 (2010). [CrossRef]

], with

F_{a} = G_{1} {(G_{1}^{'})}^{*} / (B_{4} B_{5}^{2})

and the nonlinear cross-Kerr refractive index

n_{2}^{} \propto Re (- i μ G_{F 1} F_{a}^{2} / h)

. When

Δ_{2}

is scanned in the experiment, the intensity

I_{1}

of the laser beam

E_{1}^{'}

and

n_{2}

almost stays constant for different detuning

Δ_{2}

. So ϕ is primarily determined by

I_{F 1}^{'}

. When the suppression condition (

Δ_{1} - Δ_{2} = 0

) is satisfied, and the intensity of FWM signal

I_{F 1}^{'}

reaches its minimum, the spatial splitting will become stronger as shown in Fig. 5(a) (the suppression positions located at

Δ_{2} \approx - 12 GHz

). While in the enhancement condition with the larger

I_{F 1}^{'}

, ϕ is decreased, and therefore the splitting is weakened correspondingly, as shown in Fig. 5(a), where the enhancement condition is located at

Δ_{2} \approx - 28.3 GHz

Fig. 5 (Color online) (a) DFWM signal

E_{F 1}

images when

Δ_{1} =

-69.1, −55.5, −38.7GHz. (b) DFWM signal images when

Δ_{1} =

-55.5, −38.7, −19.2GHz.

Specially, we observed the y-direction spatial splitting images of the DFWM signal

E_{F 1}

[Fig. 5(b)] by carefully arranging laser beams

k_{1}

and

k_{1}^{'}

. In the experiment, the beams

E_{1}

and

E_{1}^{'}

are deliberately aligned in y-z plane with an angle θ (

\sim {0.05}^{\circ}

) to induce a grating in the same plane with the fringe spacing

Λ = λ_{1} / θ

. Because θ is far less than the angle of

E_{1}

and

E_{1}^{'}

in the x-z plane, Λ is big enough for observing the splitting caused by the induced grating. Furthermore when

E_{1}

and

E_{1}^{'}

are set in the middle of the oven,

E_{F 1}

and

E_{1}^{'}

overlap in y-direction due to the phase matching condition. As a result, the splitting of

E_{F 1}

in x-direction due to the nonlinear cross-Kerr effect from

E_{1}^{'}

disappears simultaneously. Because Λ remains nearly the same for the changeless θ and

λ_{1}

, a larger spot of the

E_{F 1}

beam with larger intensity will be split to more parts. In Fig. 5(b), the field

E_{3}

is stronger than that in Fig. 5(a), which leads to stronger FWM signals passing through the grating in y-direction. So we can easily obtain the splitting in y-direction. Moreover, in the enhanced position, the profile of the FWM signal become larger, and more split parts induced by the grating can be obtained. Here, Figs. 5(b1)-5(b3) show the experimental spots corresponding to the curves in Figs. 4(c1)-4(c3). However, the effects of suppression and enhancement of the DFWM signal

E_{F 1}

are much worse due to the special spatial alignment of the laser beams, as shown in Fig. 4(c) compared to that in Fig. 4(a). In Fig. 4(a), the

E_{1}

E_{1}^{'}

and

E_{2}

E_{2}^{'}

are all set at the back of the heat oven. But in Fig. 4(c), only

E_{1}

and

E_{1}^{'}

are deliberately moved to the middle of the oven to demonstrate the splitting of

E_{F 1}

in the y-direction. So the dressing effect on

E_{F 1}

E_{2}^{'}

in Fig. 4(c) appears worse than that in Fig. 4(a).

In order to verify the cascade dressing interaction and two dispersion centers of FWM image. The dresses field

E_{2}^{'}

is tuned to the line center (568.8 nm) of the

| 1 >

| 2 >

(

4 D_{3 / 2, 5 / 2}

) transition, and a ladder type three-level atomic system forms, as shown in Fig. 1(b). With the dressed perturbation chains, we can obtain

I_{E_{F 1}}^{″} \propto | ρ_{E_{F 1}}^{″ (3)} |^{2}

, where

ρ_{E_{F 1}}^{″ (3)} = - i G_{3} G_{1} {(G_{1}^{'})}^{*} / (B_{6} B_{7}^{2})

, with

d_{8} = Γ_{00} + i (Δ_{1} + Δ_{2})

d_{9} = Γ_{21} + i Δ_{2}

d_{10} = Γ_{11} + A_{9}

d_{11} = d_{1} + A_{7}

A_{7} = G_{2}^{2} / d_{8}

A_{8} = G_{1}^{2} / d_{11}

A_{9} = G_{2}^{2} / d_{9}

A_{10} = G_{1}^{2} / d_{10}

B_{6} = Γ_{00} + A_{8}

and

B_{7} = d_{1} + A_{7} + A_{10}

We repeated above the experiment with the same experimental conditions [the data points in Fig. 4(a)], and obtained the results as shown in Fig. 6(a) . Comparing the results in Fig. 6(a) with those in Fig. 4(a), we can obtain the similar observations of suppression and enhancement of the DFWM signal

E_{F 1}

, except the shapes of partial suppressions/ enhancement. For instance, the curve of Fig. 4(a2) shows first suppression, and next enhancement, is different from the curve of Fig. 6(a2), which shows first enhancement, and next suppression. The reason for this contrast is the difference of levels structure. As

Δ_{1}

is set near the resonance point (

Δ_{1} = - 55.8 GHz

), the new dressed-state

| G_{1} - - >

moves from upper to lower as

Δ_{2}

is scanned from positive to zero [Fig. 4(d2)]. So the DFWM signal

E_{F 1}

is first enhanced as the condition

Δ_{1} + Δ_{2} = - G_{2}^{}

is satisfied, and then suppressed (the suppression condition

Δ_{1} + Δ_{2} = 0

), as shown in the curve of Fig. 6(a2). Simultaneously, we obtain the corresponding suppressions and enhancements of x direction spatial splitting of DFWM signal

E_{F 1}

images, as shown in Fig. 6(c). Here Figs. 6(c1)–6(c9) show the experimental spots corresponding to the curves in Figs. 6(a1)–6(a9).

Fig. 6 (a) Measured suppression and enhancement of DFWM signal

E_{F 1}

versus

Δ_{2}

for different

Δ_{1}

values.

Δ_{1} =

-70.5, −55.8, −32.7, −15.4, 0, 16.5, 32.5, 48.2 and 72.1 GHz, respectively. The dashed curve is the double-peak DFWM signal

E_{F 1}

versus

Δ_{1}

. (b) Theoretical plots corresponding to the experimental parameters in (a). (c1)-(c9) DFWM signal

E_{F 1}

images. The condition here is the same as that in (a).

4. Conclusion

In conclusion, we have experimentally observed the suppression and enhancement of the spatial FWM signal by the controlled cascade interaction of additional dressing fields, and the corresponding controlled spatial splitting of FWM signal caused by the enhanced cross-Kerr nonlinearity due to atomic coherence in two- and three-level atomic systems. In addition, we report the interplay between the two coexisting FWM signals, which can be tuned to overlap or separate by varying frequency detunings. Such controllable FWM processes can have important applications in wavelength conversion for spatial signal processing and optical communication.

Acknowledgments

This work was supported by NSFC (10974151, 61078002, 61078020), NCET (08-0431), RFDP (20100201120031), 2009xjtujc08, xjj20100100, xjj20100151.

References and links

1.	S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–9999 (1997). [CrossRef]
2.	P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, and P. Kumar, “Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium,” Opt. Lett. 20(9), 982–984 (1995). [CrossRef] [PubMed]
3.	B. Lü, W. H. Burkett, and M. Xiao, “Nondegenerate four-wave mixing in a double-Lambda system under the influence of coherent population trapping,” Opt. Lett. 23(10), 804–806 (1998). [CrossRef] [PubMed]
4.	H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80(2), 023820 (2009). [CrossRef]
5.	S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98(5), 053601 (2007). [CrossRef] [PubMed]
6.	Y. Zhang, B. Anderson, A. W. Brown, and M. Xiao, “Competition between two four-wave mixing channels via atomic coherence,” Appl. Phys. Lett. 91(6), 061113 (2007). [CrossRef]
7.	M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]
8.	M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A 64(1), 013412 (2001). [CrossRef]
9.	Z. Q. Nie, H. B. Zheng, P. Z. Li, Y. M. Yang, Y. P. Zhang, and M. Xiao, “Interacting multi-wave mixing in a five-level atomic system,” Phys. Rev. A 77(6), 063829 (2008). [CrossRef]
10.	Y. P. Zhang, Z. Q. Nie, Z. G. Wang, C. B. Li, F. Wen, and M. Xiao, “Evidence of Autler-Townes splitting in high-order nonlinear processes,” Opt. Lett. 35(20), 3420–3422 (2010). [CrossRef] [PubMed]
11.	C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. 95(4), 041103 (2009). [CrossRef]
12.	G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. 64(21), 2487–2490 (1990). [CrossRef] [PubMed]
13.	R. S. Bennink, V. Wong, A. M. Marino, D. L. Aronstein, R. W. Boyd, C. R. Stroud Jr, S. Lukishova, and D. J. Gauthier, “Honeycomb pattern formation by laser-beam filamentation in atomic sodium vapor,” Phys. Rev. Lett. 88(11), 113901 (2002). [CrossRef] [PubMed]
14.	A. J. Stentz, M. Kauranen, J. J. Maki, G. P. Agrawal, and R. W. Boyd, “Induced focusing and spatial wave breaking from cross-phase modulation in a self-defocusing medium,” Opt. Lett. 17(1), 19–21 (1992). [CrossRef] [PubMed]
15.	H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87(7), 073601 (2001). [CrossRef] [PubMed]
16.	W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous interaction of spatial solitons in photorefractive media,” Phys. Rev. Lett. 80(15), 3240–3243 (1998). [CrossRef]
17.	Y. P. Zhang, Z. G. Wang, H. B. Zheng, C. Z. Yuan, C. B. Li, K. Q. Lu, and M. Xiao, “Four-wave-mixing gap solitons,” Phys. Rev. A 82(5), 053837 (2010). [CrossRef]
18.	P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, “Storage and retrieval of multimode transverse images in hot atomic Rubidium vapor,” Phys. Rev. Lett. 100(12), 123903 (2008). [CrossRef] [PubMed]
19.	V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008). [CrossRef] [PubMed]
20.	W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. L. Luther-Davies, “Observation of dipole-mode vector solitons,” Phys. Rev. Lett. 85(7), 1424–1427 (2000). [CrossRef] [PubMed]
21.	Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four- wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102(1), 013601 (2009). [CrossRef] [PubMed]

OCIS Codes
(190.3270) Nonlinear optics : Kerr effect
(190.4180) Nonlinear optics : Multiphoton processes
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(270.1670) Quantum optics : Coherent optical effects
(300.2570) Spectroscopy : Four-wave mixing

ToC Category:
Nonlinear Optics

History
Original Manuscript: April 11, 2011
Revised Manuscript: May 21, 2011
Manuscript Accepted: June 19, 2011
Published: June 30, 2011

Citation
Changbiao Li, Yanpeng Zhang, Huaibin Zheng, Zhiguo Wang, Haixia Chen, Suling Sang, Ruyi Zhang, Zhenkun Wu, Liang Li, and Peiying Li, "Controlling cascade dressing interaction of four-wave mixing image," Opt. Express 19, 13675-13685 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13675

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References

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–9999 (1997). [CrossRef]
P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, and P. Kumar, “Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium,” Opt. Lett. 20(9), 982–984 (1995). [CrossRef] [PubMed]
B. Lü, W. H. Burkett, and M. Xiao, “Nondegenerate four-wave mixing in a double-Lambda system under the influence of coherent population trapping,” Opt. Lett. 23(10), 804–806 (1998). [CrossRef] [PubMed]
H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80(2), 023820 (2009). [CrossRef]
S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98(5), 053601 (2007). [CrossRef] [PubMed]
Y. Zhang, B. Anderson, A. W. Brown, and M. Xiao, “Competition between two four-wave mixing channels via atomic coherence,” Appl. Phys. Lett. 91(6), 061113 (2007). [CrossRef]
M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]
M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A 64(1), 013412 (2001). [CrossRef]
Z. Q. Nie, H. B. Zheng, P. Z. Li, Y. M. Yang, Y. P. Zhang, and M. Xiao, “Interacting multi-wave mixing in a five-level atomic system,” Phys. Rev. A 77(6), 063829 (2008). [CrossRef]
Y. P. Zhang, Z. Q. Nie, Z. G. Wang, C. B. Li, F. Wen, and M. Xiao, “Evidence of Autler-Townes splitting in high-order nonlinear processes,” Opt. Lett. 35(20), 3420–3422 (2010). [CrossRef] [PubMed]
C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. 95(4), 041103 (2009). [CrossRef]
G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. 64(21), 2487–2490 (1990). [CrossRef] [PubMed]
R. S. Bennink, V. Wong, A. M. Marino, D. L. Aronstein, R. W. Boyd, C. R. Stroud, S. Lukishova, and D. J. Gauthier, “Honeycomb pattern formation by laser-beam filamentation in atomic sodium vapor,” Phys. Rev. Lett. 88(11), 113901 (2002). [CrossRef] [PubMed]
A. J. Stentz, M. Kauranen, J. J. Maki, G. P. Agrawal, and R. W. Boyd, “Induced focusing and spatial wave breaking from cross-phase modulation in a self-defocusing medium,” Opt. Lett. 17(1), 19–21 (1992). [CrossRef] [PubMed]
H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87(7), 073601 (2001). [CrossRef] [PubMed]
W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous interaction of spatial solitons in photorefractive media,” Phys. Rev. Lett. 80(15), 3240–3243 (1998). [CrossRef]
Y. P. Zhang, Z. G. Wang, H. B. Zheng, C. Z. Yuan, C. B. Li, K. Q. Lu, and M. Xiao, “Four-wave-mixing gap solitons,” Phys. Rev. A 82(5), 053837 (2010). [CrossRef]
P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, “Storage and retrieval of multimode transverse images in hot atomic Rubidium vapor,” Phys. Rev. Lett. 100(12), 123903 (2008). [CrossRef] [PubMed]
V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008). [CrossRef] [PubMed]
W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. L. Luther-Davies, “Observation of dipole-mode vector solitons,” Phys. Rev. Lett. 85(7), 1424–1427 (2000). [CrossRef] [PubMed]
Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four- wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102(1), 013601 (2009). [CrossRef] [PubMed]

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